Weapons permits (waffenrechtliche Erlaubnisse) can be issued to the home-based foreign members of diplomatic missions and career consular posts upon application (free of charge if reciprocity has been agreed). The authority responsible for this is the

Bundesverwaltungsamt (Federal Office of Administration)
Referat S II 6,50728 Köln
E-Mail: waffenrecht@bva.bund.de

https://www.bva.bund.de/DE/Services/Buerger/Ausweis-Dokumente-Recht/Waffenrecht/Einzelerlaubnis/_documents/Antrag/Antrag_Erlaubnis.html

For members of diplomatic missions, the application (Annexes WA1 and WA2, plus all documents listed there) is submitted via Division 703 at the Federal Foreign Office, and for members of career consular posts, it is forwarded to the Federal Office of Administration by the competent Staatskanzlei or Senatskanzlei. The Federal Office of Administration issues the weapons permits and sends them directly to the mission or post.

More information can be found in Circular Note 04/2019 of 8 February 2019 – Ref. 701-1-701 AM 15 (Weapons permits) including the accompanying information sheet of the Federal Office of Administration.